Let $\Sigma$ be a fixed signature, and $\A$ and $\B$ be two structures for $\Sigma$ The interesting functions from $\A$ to $\B$ are the ones that preserve the structure.

A function $f\colon \A \to \B$ is said to be a homomorphism (or simply morphism) if and only if:

1. For every constant symbol $c$ of $\Sigma$ $f(c^\A)=c^\B$
2. For every natural number $n$ and every $n$ ary function symbol $F$ of $\Sigma$ $$ f(F^\A(a_1,...,a_n))=F^\B(f(a_1),...,f(a_n)). $$
3. For every natural number $n$ and every $n$ ary relation symbol $R$ of $\Sigma$ $$ R^\A(a_1, \ldots ,a_n) \Implies R^\B(f(a_1), \ldots,f(a_n)). $$

Homomorphisms with various additional properties have special names:

• An injective homomorphism is called a monomorphism.
• A surjective homomorphism is called an epimorphism.
• A bijective homomorphism is called a bimorphism.
• An injective homomorphism $f$ is called an embedding if, for every natural number $n$ and every $n$ ary relation symbol $R$ of $\Sigma$ $$ R^\B(f(a_1), \ldots,f(a_n)) \Implies R^\A(a_1, \ldots ,a_n), $$ the converse of condition 3 above, holds.
• A surjective embedding is called an isomorphism.
• A homomorphism from a structure to itself (e.g., $f\colon \A \to \A$ is called an endomorphism.
• An isomorphism from a structure to itself is called an automorphism.